Orthogonal frequency division multiplexing (OFDM) is a wideband modulation radio frequency data transmission mode in which a frequency bandwidth allocated for a communication session is divided into a plurality of narrow band frequency sub-bandwidths. Each sub-bandwidth includes a radio frequency (RF) subcarrier. Subcarriers in different sub-channels are mathematically orthogonal to each other.
The orthogonality of the subcarriers allows individual spectrums of the subcarriers to be overlapped without inter-carrier-interference (ICI). Since a frequency bandwidth is divided into a plurality of orthogonal sub-bandwidths, OFDM allows a high data transmission rate and high bandwidth use efficiency.
OFDM is a multi-carrier modulation scheme of converting data to be transmitted into an M-ary quadrature amplitude modulation (QAM) complex symbol, converting a complex symbol sequence into a plurality of parallel complex symbols through serial-to-parallel conversion, and performing rectangular pulse shaping and subcarrier modulation of each parallel complex symbol. In the multi-carrier modulation, a frequency interval between subcarriers is set such that all of the subcarrier modulated parallel complex symbols are orthogonal to each other.
In a case where an M-ary QAM modulation signal is transmitted through a wireless fading channel without using OFDM, if a channel delay spread caused by multipath delay is greater than a symbol period of the modulation signal, inter-symbol interference (ISI) occurs and hinders a receiver from correctly recovering the signal. For this reason, an equalizer that compensates for a random delay spread is typically used in the receiver. However, the equalizer is very complicated to implement and the performance of the receiver is greatly degraded due to input noise.
On the other hand, when OFDM is used, the symbol period of each parallel complex symbol can be expanded to be much longer than the channel delay spread. Accordingly, ISI can be decreased. In particular, when a guard interval is set to be longer than the delay spread, ISI can be completely eliminated. In addition, it is not necessary to implement the equalizer that compensates for a random delay spread caused by multipath delay. Accordingly, OFDM is very effective in data transmission through a wireless fading channel and has been thus adopted as a standard transmission mode for terrestrial digital television (DTV) and audio broadcasting systems in Europe.
Especially, in time-domain synchronous (TDS) OFDM systems, a pseudo-noise (PN) sequence rather than a cyclic prefix (CP) is inserted as a guard interval between inverse discrete Fourier transformed data blocks. Since the PN sequence is also used as a training symbol in an OFDM receiver, higher spectrum efficiency can be obtained in OFDM systems using the PN sequence than in OFDM systems using the CP.
Meanwhile, ICI occurring in a mobile channel causes serious degradation of system performance. The characteristic narrow bandwidth of a subcarrier in an OFDM system is advantageous in overcoming frequency selectivity caused by multipath delay spread but is relatively more sensitive to time selectivity caused by rapid time variation in a mobile channel due to the orthogonality of subcarriers. The time variation may affect the orthogonality of OFDM subcarriers and thus cause ICI.
When a complex modulation symbol vector in a frequency domain is expressed in the form X=[X0 X1 . . . XNc-1]T, a baseband complex signal “x” in a time domain may be expressed by Equation (1):x=[x0 x1 . . . xNc-1]T=FHX  (1)where a Fourier transform matrix
      F    =                  (                  F          nk                )                              N          c                ×                  N          c                      ,          ⁢            F      nk        =                  1                              N            c                              ⁢              ⅇ                              -            ⅈ                    ⁢                                    2              ⁢              π                                      N              c                                ⁢          nk                      ,Nc is the size of fast Fourier transform (FFT)/inverse FFT (IFFT), and [ ]T denotes a matrix transpose operation and [ ]H denotes a conjugate transpose operation.
In addition a channel impulse response (CIR) in a mobile wireless channel may be expressed by Equation (2):
                              h          ⁡                      (                          t              ,              τ                        )                          =                              ∑                          k              =              0                                      L              -              1                                ⁢                                                    γ                k                            ⁡                              (                t                )                                      ⁢                          δ              ⁡                              (                                  τ                  -                                      τ                    k                                                  )                                                                        (        2        )            where τk is a time delay on a k-th path among L paths and γk(t) is a complex gain. Accordingly, a discrete form corresponding to h(t,τ) may be expressed by h[m,l]=h(t=mTs,τ=ITs) where Ts is a sampling period.
When a received signal is represented by y=[y0 y1 . . . yNc-1]T, a convolution process may be expressed by Equation (3):
                              y          i                =                                                            h                ⁡                                  [                                      i                    ,                    l                                    ]                                            *                              x                l                                      +                          z              i                                =                                                    ∑                                  l                  =                  0                                                  L                  -                  1                                            ⁢                                                          ⁢                                                h                  ⁡                                      [                                          i                      ,                      l                                        ]                                                  ⁢                                  x                                      i                    -                    l                                                                        +                          z              i                                                          (        3        )            where “*” indicates a convolution operation, zis a zero mean, and σ2 is a Gaussian random variance. The convolution process may be expressed by Equations (4) and (5) in a vector form based on Equation (3).
                              y          0                =                                                            [                                                      h                    ⁡                                          [                                              0                        ,                        0                                            ]                                                        ⁢                  0                  ⁢                                                                          ⁢                  …                  ⁢                                                                          ⁢                  0                  ⁢                                                                          ⁢                                      h                    ⁡                                          [                                              0                        ,                                                  L                          -                          1                                                                    ]                                                        ⁢                                                                          ⁢                  …                  ⁢                                                                          ⁢                                      h                    ⁡                                          [                                              0                        ,                        1                                            ]                                                                      ]                            ⁡                              [                                                                                                    x                        0                                                                                                                                                x                        1                                                                                                                        …                                                                                                                          x                                                                              N                            c                                                    -                          1                                                                                                                    ]                                      +                          z              0                                =                                                    Θ                                  (                  0                  )                                            ⁢              x                        +                          z              0                                                          (        4        )                                          y          1                =                                                            [                                                      h                    ⁡                                          [                                              1                        ,                        1                                            ]                                                        ⁢                                      h                    ⁡                                          [                                              1                        ,                        0                                            ]                                                        ⁢                                                                          ⁢                  …                  ⁢                                                                          ⁢                  0                  ⁢                                                                          ⁢                  0                  ⁢                                                                          ⁢                  …                  ⁢                                                                          ⁢                                      h                    ⁡                                          [                                              1                        ,                        2                                            ]                                                                      ]                            ⁡                              [                                                                                                    x                        0                                                                                                                                                x                        1                                                                                                                        …                                                                                                                          x                                                                              N                            c                                                    -                          1                                                                                                                    ]                                      +                          z              1                                =                                                    Θ                                  (                  1                  )                                            ⁢              x                        +                          z              1                                                          (        5        )            where Θ(0)=[h[0,0] 0 . . . 0 h[0,L−1] . . . h[0,1]] and Θ(1)=[h[1,1] h[1,0] . . . 0 0 . . . h[1,2]] indicates cyclical shift of time-variant Θ(0).
Accordingly, Equation (3) may be rewritten as Equation (6) in a matrix form:y=AFHX+z  (6)where A=[Θ(0) Θ(1) . . . Θ(Nc−1)]T and z=[z0 z1 . . . zNc-1]T. When a coefficient matrix in Equation (6) is expressed by Equation (7), Equation (8) is obtained.E=AFH=(Epq)Nc×Nc  (7)
                              E          pq                =                              1                                          N                c                                              ⁢                                    ∑                              l                =                0                                            L                -                1                                      ⁢                                                  ⁢                                          h                ⁡                                  [                                      p                    ,                    l                                    ]                                            ⁢                              ⅇ                                  j                  ⁢                                                            2                      ⁢                      π                                                              N                      c                                                        ⁢                                      (                                          p                      -                      l                                        )                                    ⁢                  q                                                                                        (        8        )            Here, since a guard interval is considered in computation, linear convolution is equivalent to cyclic convolution.
When FFT is performed according to Equation (6), a frequency domain demodulation signal can be obtained by Equation (9):Y=Fy=FAFHX+Z=GX+Z  (9)where Z=Fz and a gain matrix is expressed by Equation (10):G=FAFH=(Gpq)Nc×Nc  (10)
Accordingly, Equation (11) can be obtained:
                              G          pq                =                              1                                          N                c                                              ⁢                                    ∑                              m                =                0                                                              N                  c                                -                1                                      ⁢                                                  ⁢                                          ∑                                  l                  =                  0                                                  L                  -                  1                                            ⁢                                                          ⁢                                                h                  ⁡                                      [                                          m                      ,                      l                                        ]                                                  ⁢                                                      ⅇ                                                                  -                        j                                            ⁢                                                                        2                          ⁢                          π                                                                          N                          c                                                                    ⁢                                              (                                                  p                          -                          l                                                )                                            ⁢                      q                                                        .                                                                                        (        11        )            
When Y=[Y0 Y1 . . . YNc-1]T, Equation (12) is obtained:
                              Y          k                =                                            G              kk                        ⁢                          X              k                                +                                                    ∑                                                      q                    =                    0                                    ,                                      q                    ≠                    k                                                                                        N                    c                                    -                  1                                            ⁢                                                          ⁢                                                G                  kq                                ⁢                                  X                  q                                                                    ︸                              ICI                ⁢                                                                  ⁢                term                                              +                                    Z              k                        .                                              (        12        )            
When a channel is time-invariant, that is; h[m,l]=h[l] in Equation (12), the ICI term becomes 0 in Equation (12) and frequency-selectivity fading can be eliminated using a one-tap equalizer. Conversely, ICI occurs in a mobile channel having a time-variant feature. Influences of ICI on OFDM are introduced by M. Russell and G. L. Stuber [“Interchannel Interference Analysis of OFDM in a Mobile Environment,” in Proc. VTC'95, vol. 2, pp. 820-824, July 1995] which is incorporated herein by reference. Accordingly, an equalization method for mitigating ICI is desired.
Meanwhile, it can be inferred from Equation (9) that a signal obtained after equalization is expressed by Equation (13):{circumflex over (X)}=G−1Y  (13)where ( )−1 is an inverse matrix operation. In other words, an equalizer should perform the inverse matrix operation in order to perform equalization. Here, since complexity of the inverse matrix operation is O(N3c) the equalization is very impractical when Nc is large.